To illustrate further the difference between the 2-D and the 3-D solutions and to verify the applicability of the simplified inversion procedure, three actual sites A (soil deposit), B and C (pavements) were studied. For each site, the
experimental dispersion curve was obtained using the SASW method. Approximate soil properties were computed using the simplified procedure and iterative analyses assuming a profile, computing the theoretical dispersion curves,
comparing them to the experimental data, and modifying the assumed properties to improve the fit.
For a soil deposit with stiffness gradually increasing with depth like site A, a set of properties which gave an excellent agreement with the experimental dispersion curve could be easily obtained with very few iterations. Fig. 6a shows the soil properties obtained by the simplified and the analytical procedures. The agreement is very good up to 10 or 12m but deteriorates for larger depths. Fig. 6b shows the theoretical dispersion curves corresponding to the soil profile obtained with the simplified inversion procedure. The agreement with the experimental data is very good up to a wavelength of approximately 30 m, but the theoretical results are noticeably smaller than the experimental ones beyond that point. Fig. 6c compares the experimental dispersion curves to the 2-D and 3-D theoretical curves corresponding to the fitted profile. The 2-D solution is a smooth curve which matches very well the experimental data except for the range of wavelengths between 35 and 50m. In this range, the experimental data show a sharp change in slope instead of the smooth behavior of the 2-D solution. The 3-D solution, on the other hand, reproduces very well this abrupt change in slope.
Site B represents an asphalt concrete pavement. Fig. 7a shows the material properties obtained with the simplified and the iterative procedure. The two profiles have now some marked differences. Fig. 7b shows the theoretical dispersion curves corresponding to the simplified profile. It should be noticed that the results of the 2-D analysis with the first mode appear to be in better agreement with the experimental data for wavelengths up to 3 m, although they are in fact less accurate. The phase velocities obtained with the 3-D solution are larger than the experimental data. Fig. 7c shows the corresponding results for the fitted profile. Even in this case for the number of iteration performed, the agreement between the theoretical dispersion curves and the experimental data is not perfect particularly for wavelengths between 1.5 and 3m. It is however much better than that of Fig. 7b.
Similar results for the material profile at site C are shown in Fig. 8a. It is apparent that the approximate profile obtained from the simplified inversion is significantly different from the fitted one for depths below 0.5m. Differences between the dispersion curves of the approximate profile and the experimental data as shown in Fig. 8b are viewed at
wavelengths from .5m to 1.5m. Dispersion results for the profile with analytical fitting are shown in Fig. 8c. The 3-D solution reproduces very well the experimental data. The use of the first mode in the 2-D solution is again inappropriate for the this site.
CONCLUSIONS
The results of the cases shown here and a large number of other studies conducted in recent years indicate that the simplified inversion procedure based on assigning a Rayleigh wave velocity at each depth equal to the phase velocity for a wavelength of 3 times that depth will produce very good results when dealing with a soil profile where the properties increase smoothly with depth. When there are layers of soil with very abrupt and marked changes in properties or when the properties decrease with depth, as in the case of pavements, the procedure can be used to obtain a preliminary estimate but must be refined through a series of analyses.
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The use of the 2-D solution with the dispersion curve corresponding to the first mode of propagation (smallest eigenvalue) is reasonable for soil deposits with gradual variation of properties but cannot reflect sudden jumps and discontinuities in the slope of the curve caused by wave reflections or refractions. For more complicated soil profiles and most pavement systems, the use of the more accurate three dimensional solution is recommended since this approach can reproduce the true wave propagation phenomena involved in the test. Even so some smoothing of the experimental data, which is not without error or noise, and even of the theoretical curves, may be necessary to get good fits with a
reasonable amount of work. This is due to the fact that the results of the 3-D analyses depend on the actual position of the points at which the phases are computed. Varying these points will introduce modifications in the dispersion curves. To obtain a reasonable agreement, the position of the receivers should thus be similar in the field and in the analyses.
REFERENCES
1. DEGEBO,“Deutsche Gesellschaft fur Bodenmechanik,” Vol. 4, Springer, Berlin, 1938.
2. Bergstorm, S.G. and Linderholm, S.,“Dynamic Method att Utrona Ultiga Marklagers Genomsnittliga Elasticitetsegens Kaper,” Handlinger No. 7, Svenska Forsknings-Institutet for Cement och Betong Vid. Kungl. Tekniska, Hogskolan, 1946.
3. Van der Poel, C., “Dynamic Testing of Road Construction,” Journal of Applied Chemistry, Vol. 1, 1951, pp. 281–290. 4. Nijboer, L.W. and Van der Poel, C.,“A Study of Vibration Phenomena in Asphaltic Road Construction,” Proceedings, Association of Asphaltic Pavement Technology, 1953, pp. 197–231.
5. Jones, R., “In-Situ Measurement of the Dynamic Properties of Soil by Vibration Methods,” Geotechnique, Vol. 8, No. 1, March, 1958, pp. 1–21.
6. Jones, R.,“Surface Wave Technique for Measuring the Elastic Properties and Thickness of Roads: Theoretical Developement,” British Journal of Applied Physics, Vol. 13, 1962, pp. 21–29.
7. Heukelom, W. and Foster, C.R., “Dynamic Testing of Pavements,” Journal of Soil Mech. and Found. Div., Proc. ASCE, Vol. 86, No. SM1, Part 1, February, 1960.
8. Ballard, R.F., “Determination of Soil Shear Moduli al Depth by In-Situ Vibratory Techniques,” Miscellaneous Paper No. 4–691, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1964.
9. Fry, Z.B.,“Dynamic Soil Investigations Project Buggy, Buckboard Mesa Neveda Test Site, Mercury, Neveda,” Miscellaneous Paper No. 4–666, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1965.
10. Heisey, J.S., Stokoc, K.H., II, and Meyer, A.H., “Moduli of pavement Systems from Spectral Analysis of Surface Waves,” Transportation Research Record, No. 852, Washington, D.C., January, 1982.
11. Slokoe, K.H., II, and Nazarian, S., “Effectiveness of Ground Improvement from Spectral Analysis of Surface Waves,” Proceedings, 8th European Conference on Soil Mechanics and Foundation Engineerin, Helsinki, Finland, May 1983.
12. Shao, K.-Y., “Dynamic Interpretation of Dynaflect, Falling Weight Deflectometer and Spectral Analysis of Surface Waves Tests on Pavement System”, Ph.D. Dissertation, The University of Texas al Austin, December 1985.
13. Sanchez-Salinero, I.,“Analytical Investigation of Seismic Methods Used for Engineering Applications”, Ph.D. Dissertation, The University of Tcxas at Austin, May 1987.
14. Sheu, J.C., “Applications and Limitations of the Spectral-Analysis-of-Surface-Waves Method”, Ph.D. Dissertation, The University of Texas at Austin, August 1987
15. Rix, G.J., “Experimental Study of Factors Affecting the Spectral-Analysis-of-Surface-Waves Methods, Ph.D. Dissertation, The University of Texas at Austin, August 1988.
16. Roesset, J.M., Chang, D.-W., Stokoe, K.H., II and Aouad, M., “Modulus and Thickness of the Pavement Surface Layer from SASW Tests,” Transportation Research Record, Vol. 1260., January 1990, pp. 53–63.
17. Kang, Y.V., “The Effect of Finite Width on Dynamic Deflections of Pavements”, Ph.D. Dissertation, The University of Texas at Austin, May 1990.
18. Vrettos, C. and Prange, B.,“Evaluation of In Situ Effective Shear Modulus from Dispersion Measurements,” Journal of Geotechnical Engineering, Vol. 116, No. 10, October, 1990, pp. 1581–1585.
19. Thomson, W.T., “Transmission of Elastic Waves Through a Stratified Soil Medium,” Journal of Applied Physics, Vol. 21, February 1950., pp. 89–93.
20. Haskell, N.A.,“The Dispersion of Surface Waves on Multilayer Media,” Bulletin of the Seismological Society of America, Vol. 43, 1953, pp. 17–34.
21. Kausel, E. and Roesset, J.M., “Stiffness Matrices for Layered Soils,” Bulletin of the Seismological Society of America, Vol. 71, 1981, pp. 1743–1761.
22. Nazarian, S., “In Situ Determination of Elastic Moduli of Soil Deposits and Pavement Systems by Spectral-Analysis- of-Surface-Waves Method”, Ph.D. Dissertation, The University of Texas at Austin, August 1984.
23. Waas, G., “Linear Two-Dimensional Analysis of Soil Dynamics Problems in Semi-Infinite Layered Media,” Ph.D. Dissertation, The University of California at Berkeley, 1972.
24. Kausel, E., “Forced Vibration of Circular Foundations on Layered Media”, Research Report R74–11, Department of Civil Engineering, M.I.T., 1974.
25. Kausel, E., “An Explicit Solution for the Green Functions for Dynamic Loads in Layered Media”, Research Report S81–13, Department of Civil Engineering, M.I.T., 1981.
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Figure 1 General configuration of equipment used in SASW test
Figure 2 Approximate distribution of vertical particle motion with depth subjected to various excitation frequencies (or wavelengths)
Figure 3 Composite dispersion curve from data of individual receiver spacing
Figure 4 Artificial profile No.1, a) material property, b) 2-D and 3-D dispersion curves, c) comparison of profile and solution from simplified inversion
Figure 5 Artificial profile No.2, a) material property, b) 2-D and 3-D dispersion curves, c) comparison of profile and solution from simplified inversion
Figure 6 Site A, a) approximate profile and computed Profile, b) dispersion curves of approximate profile, c) dispersion curves of computed profile
curves of computed profile
Figure 8 Site C, a) approximate profile and computed profile, b) dispersion curves of approximate profile, c) dispersion curves of computed profile
Inversion of Rayleigh Wave Dispersion Curve for SASW Test
N.Gucunski (*), R.D.Woods (**)
(*) Department of Civil & Environmental Engineering, Rutgers University, P.O. Box 909, Piscataway, NJ 08855–0909, U.S.A.
(**) Department of Civil Engineering, The University of Michigan, Ann Arbor, MI 48109–2125, U.S.A.
INTRODUCTION
The Spectral-Analysis-of-Surface-Waves method is a seismic technique for measuring in situ elastic moduli and thicknesses of layered systems, like soils and pavements. The advantages of the method are: it is performed from the surface and therefore does not require boreholes, it is nondestructive, it is in most cases highly accurate, and it utilizies a simple procedure and test setup with the prospect of being fully automized. The major deficiency of the method at this moment is the process of inversion of the dispersion curve.
The inversion process is a simple task and provides reliable results in cases of regular soil profiles where the shear wave velocity increases with depth. Experience which the authors have from SASW measurements in which lower velocity layers were found below higher velocity layers by a complementary crosshole test, is that the inversion process can be an ambiguous process.
Results of the study of Rayleigh wave dispersion in soil profiles where a softer layer is trapped between a harder surface layer and a harder half-space have indicated (Gucunski and Woods, 1991) that the uniqueness of a derived soil profile is attributed to a strong influence of higher Rayleigh modes on the overall wave propagation pattern. This paper presents results on Rayleigh wave dispersion for several cases of soil layering which can be characterized as irregular patterns of soil stratification. The cases include:
• A softer layer trapped between a harder surface layer and a half-space, • A hard surface layer, and
• A harder layer trapped between a softer surface layer and a half-space.
The goal of the study is to explore alternative ways for the improvement of the inversion process and to provide guidelines for identification and interpretation of results in the field.